On local fields invariant under the action of topological defects

TL;DR

This paper investigates the construction and classification of local operators and topological defects in rational conformal field theories, providing solutions for SU(3) WZW theory and proposing methods for general cases.

Contribution

It offers a detailed analysis of fusion rule problems related to topological defects and boundary operators, including a complete solution for SU(3) WZW theory and a framework for broader theories.

Findings

Constructed commuting boundary operators and junctions for SU(3) WZW theory.

Analyzed fusion rule restrictions for charge conjugation invariant configurations.

Proposed an approach for classifying defects in general WZW theories.

Abstract

In the context of rational conformal field theories (RCFT) we look into the problem of constructing and classifying pairs consisting of a local operator and a topological defect which commutes or anticommutes with it. We discuss the bulk and boundary versions of the problem. In the latter one considers a conformal boundary condition, a boundary operator on it and a junction with a topological defect. In the case of the charge conjugation modular invariant commuting configurations in each problem can be obtained when a certain restriction on the fusion rules in realised. We study the corresponding fusion rule problems in detail. While in the bulk case it reduces to realising the $a\times b = c$ fusion rule which was studied in arXiv:2012.14689 [hep-th], in the boundary it leads to a new type of problem. We obtain a full solution to this problem for the $\mathrm{SU(3)}$ WZW theory, thus constructing a class of commuting boundary operators and junctions in that theory, and suggest an approach to general WZW theories.